Ok, and generally speaking, a local invariant for Riemannian geometry is a mathematical object O(p) associated to a point p on a Riemannian manifold (M,g) such that if there exists f:(M,g)-->(M',g') a local isometry at p, then O(f(p))= O(p).
A local invariant of topological manifolds is dimension.
A local invariant for symplectic manifolds would be an object O(p) associated to a point p on a symplectic manifold (M,ω) such that if there exists f:(M,ω)-->(M',ω') a local symplectomorphism at p, then O(f(p))= O(p). But by Darboux's theorem, there is no local invariant that is "properly symplectic", in the sense that every symplectic invariant would also be a smooth invariant, and thus would not be helpful in classifying symplectic manifolds. for this, we can only rely on global invariant.
Thank you for this great link mma!
Question: Are there properly smooth invariants (local or global)? i.e. do we know of a way to associate an object to a smooth manifold such that this object is the same for two diffeomorphic manifolds, but not necessarily so if the smooth manifolds are simply homeomorphic?